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Whelk-Come to Mathematics: Using Rational Functions
to Investigate the behavior of Northwestern Crows
Part Three - Analyze the
Data
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The amount of work in dropping a whelk to break it
open depends on the height of the drop and the number
of times a whelk has to be dropped.
Work = Height * Number of Drops
W = H * N
To investigate the work solely as a function of the
height, a relationship between the number of drops and
the height of the drop is required.
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You may have observed that the data for (H,
N) resembles the graph of the hyperbolic
function
which has a horizontal asymptote of x
= 0 and a vertical asymptote of y = 0.
The general equation for hyperbolic graphs is:
The goal of this activity is to find a good model
for the relationship between the height of the
drop H and the number of drops N.
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For this investigation, you will need to specify a
data set to use. The last data set you have been using
may already be specified. The sample peanut data is
from the Conduct an Experiment
activity.
Reto Zach carried out an experiment of dropping whelks
of various sizes from different heights. Since northwestern
crows feed only on the largest whelk, the data from
his experiment for the large whelk are provided (See
Assignment Below).
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Graphical Analysis
Using the interactive graph, try different
values of a, b and c in the equation:
to find a graph that fits the data.
Based on your work with the interactive graph,
answer the following questions.
- How do the values of a, b and c change the
shape of the graph?
- Based on the situation, what is a reasonable
conjecture and possible explanation for the
most likely value of a?
- Using this value of a, what are values for
b and c so that the function closely models
the data?
- Which data points cause the most difficulty
regarding getting the model to fit? What explanation
can you give for this observation?
- Explain why a function of the form:
is a reasonable conjecture for the data.
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Symbolic Analysis
When ready, click on the following
button to show the transformed data and the
corresponding graph. A line of best fit is automatically
calculated for you and plotted with the data.
- Find the equation of the line for the relationship
between H and (N - 1)-1.
Rewrite this equation to express the hyperbolic
relationship between H and N.
- Compare the equation derived using this
method to the equation you found above.
- Be sure to write your equations using
the variables H and N
- Why do you think it is necessary to assume
a value for a to be able to use this
method?
- What are the horizontal and vertical asymptotes
for your equations? Do these values make sense?
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Own Your Own
Reto Zach collected data on different
sizes of whelk based on weight. The graph of his
results and the curves he sketched by hand are
shown in the following diagram.
- What do you observe about the vertical and
horizontal asymptotes for the different sizes
of whelks? Explain why the asymptotes might
differ for the different sizes of whelks.
- What are possible equations for each size
of whelk?
- Using the data for the largest whelk (See
the Data Table above), find an equation using
the methods of this investigation.
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Credits and References
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